State Vector

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This page derives the 5-dimensional state vector used throughout the Apogee simulator.

Coordinate System Choice

We model the rocket ascent in a 2D inertial plane centered at Earth's center. This planar approximation is valid for:

  • Equatorial launches (inclination ≈ 0°)
  • Short flight times (< 15 minutes)
  • Negligible Earth rotation during ascent

LaTeX Reference

This section corresponds to Section 2 of nm_final_project.tex.

State Vector Definition

The complete state of the rocket at time \(t\) is described by:

\[ \boxed{ \mathbf{y}(t) = \begin{bmatrix} r(t) \\ \lambda(t) \\ v(t) \\ \gamma(t) \\ m(t) \end{bmatrix} } \tag{Eq. 1} \]

Two-Stage Rocket Launch

Planar polar coordinates for rocket ascent

State Variables

Geocentric Radius \(r(t)\)

Definition: Distance from Earth's center to the rocket.

\[ r(t) = R_E + h(t) \]

Where: - \(R_E = 6{,}371{,}000\) m (Earth's mean radius) - \(h(t)\) = altitude above sea level [m]

Physical range: \(r \geq R_E\) (rocket is above surface)

Implementation: dynamics.py line 37

Downrange Central Angle \(\lambda(t)\)

Definition: Angular displacement measured from launch site.

\[ \lambda(t) = \frac{\text{arc length}}{r} = \frac{s}{r} \]

Physical interpretation: - \(\lambda = 0\) at launch - At orbit insertion, typically \(\lambda \approx 0.2\) rad (≈12°)

Implementation: dynamics.py line 38

Speed Magnitude \(v(t)\)

Definition: Magnitude of the velocity vector.

\[ v(t) = \|\mathbf{v}(t)\| = \sqrt{v_r^2 + v_\theta^2} \]

Where: - \(v_r = v\sin\gamma\) (radial component) - \(v_\theta = v\cos\gamma\) (tangential component)

Physical range: - Liftoff: \(v = 0\) - Orbit insertion: \(v \approx 7800\) m/s

Implementation: dynamics.py line 39

Flight-Path Angle \(\gamma(t)\)

Definition: Angle between velocity vector and local horizontal.

\[ \gamma = \arctan\left(\frac{v_r}{v_\theta}\right) \]

Rocket Forces

Flight-path angle γ measured from local horizontal

Sign convention: - \(\gamma > 0\): Ascending (gaining altitude) - \(\gamma = 0\): Horizontal flight - \(\gamma < 0\): Descending

Key values: | Phase | \(\gamma\) value | |-------|---------------| | Vertical ascent | \(\pi/2\) (90°) | | After pitchover | \(\approx 80°\) | | Orbit insertion | \(\approx 0°\) |

Implementation: dynamics.py line 40

Vehicle Mass \(m(t)\)

Definition: Total instantaneous mass of the rocket.

\[ m(t) = m_{\text{structure}} + m_{\text{propellant}}(t) + m_{\text{payload}} \]

Variable Mass

Variable mass dynamics during propulsive flight

Mass budget (Falcon 9 v1.2 FT):

Component Mass [kg]
Stage 1 propellant ~411,000
Stage 1 dry 22,000
Stage 2 propellant ~107,000
Stage 2 dry 4,000
Interstage 2,000
Payload 0-10,000
Total at liftoff ~549,000

Implementation: dynamics.py line 41, falcon9.py

Velocity Decomposition

The velocity vector in polar coordinates:

\[ \mathbf{v} = v_r \hat{e}_r + v_\theta \hat{e}_\theta \]

Relating to state variables:

\[ \begin{aligned} v_r &= v \sin\gamma = \frac{dr}{dt} \\ v_\theta &= v \cos\gamma = r\frac{d\lambda}{dt} \end{aligned} \]

This decomposition leads directly to the kinematic equations derived in Equations of Motion.

Initial Conditions

For a launch from Earth's surface:

\[ \mathbf{y}(0) = \begin{bmatrix} R_E \\ 0 \\ 0 \\ \frac{\pi}{2} \\ m_0 \end{bmatrix} \]
Variable Initial Value Meaning
\(r(0)\) \(R_E\) On Earth's surface
\(\lambda(0)\) \(0\) At launch site
\(v(0)\) \(0\) Stationary
\(\gamma(0)\) \(\pi/2\) Vertical
\(m(0)\) \(m_0\) Gross liftoff mass

Implementation: simulate.py lines 220-230

Why Polar Coordinates?

The polar representation offers several advantages:

  1. Natural gravity alignment: Gravity points radially inward (\(-\hat{e}_r\))
  2. Altitude is direct: \(h = r - R_E\)
  3. Angular momentum: \(L = r \cdot v \cos\gamma\) is simple
  4. Orbital insertion: Circular orbit has \(\gamma = 0\), \(v = \sqrt{\mu/r}\)

Code Implementation

@dataclass(frozen=True, slots=True)
class AscentConfig:
    earth: EarthParams
    mission: MissionParams
    vehicle: VehicleParams
    numerics: NumericsParams

The state vector indices are defined as constants:

_R = 0      # Geocentric radius index
_LAM = 1    # Downrange angle index
_V = 2      # Speed magnitude index
_GAMMA = 3  # Flight-path angle index
_M = 4      # Mass index

References

  1. Battin, R.H. (1999). An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series. Chapter 3 - "The Two-Body Problem".

  2. Curtis, H.D. (2013). Orbital Mechanics for Engineering Students. Butterworth-Heinemann. Chapter 2 - "The Two-Body Problem".

Next Steps

Theory Equations of Motion